Forgive me if this is incorrect because it's been a while since I've dived into this stuff also.

The diaphragm and the case are both directly connected to the static pressure, therefore in level flight the pressure is equal in both, indication 0 fpm. As you climb, the pressure in the diaphragm decreases. The static pressure inside the case also decreases, but due to the calibrated leak, the pressure decreases at a slower rate. This causes greater differential pressure in the case, compressing the diaphragm which shows a climb in the VSI.

In a descent it's exactly the opposite. The diaphragm expands due to greater static pressure compared to the gradual increase in the case. The expansion then shows a decrease on the VSI.